LMFDB - Embedded newform 4028.2.c.a.3497.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4028,2,Mod(3497,4028)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4028.3497");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4028.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.1637419342$
Analytic rank:	$0$
Dimension:	$82$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3497.3
Character	$\chi$	$=$	4028.3497
Dual form			4028.2.c.a.3497.80

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.26219i q^{3} +2.63587i q^{5} -3.36239 q^{7} -7.64188 q^{9} +O(q^{10})$	$q-3.26219i q^{3} +2.63587i q^{5} -3.36239 q^{7} -7.64188 q^{9} -3.00331 q^{11} -6.81042 q^{13} +8.59870 q^{15} -1.96976 q^{17} +1.00000i q^{19} +10.9687i q^{21} -9.19408i q^{23} -1.94780 q^{25} +15.1427i q^{27} +0.0365021 q^{29} +5.90082i q^{31} +9.79737i q^{33} -8.86281i q^{35} +5.76204 q^{37} +22.2169i q^{39} -4.04919i q^{41} +8.69437 q^{43} -20.1430i q^{45} -0.478803 q^{47} +4.30566 q^{49} +6.42575i q^{51} +(-3.57810 + 6.34012i) q^{53} -7.91633i q^{55} +3.26219 q^{57} +6.03483 q^{59} -7.29057i q^{61} +25.6950 q^{63} -17.9514i q^{65} -6.45017i q^{67} -29.9928 q^{69} -10.2064i q^{71} -5.84713i q^{73} +6.35409i q^{75} +10.0983 q^{77} +4.07524i q^{79} +26.4727 q^{81} +10.0109i q^{83} -5.19204i q^{85} -0.119077i q^{87} -16.5171 q^{89} +22.8993 q^{91} +19.2496 q^{93} -2.63587 q^{95} -3.32473 q^{97} +22.9510 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q - 8 q^{7} - 82 q^{9}+O(q^{10}) $ 82 * q - 8 * q^7 - 82 * q^9	$ 82 q - 8 q^{7} - 82 q^{9} + 4 q^{13} + 4 q^{15} - 4 q^{17} - 58 q^{25} - 16 q^{29} - 12 q^{37} - 32 q^{43} + 8 q^{47} + 98 q^{49} + 6 q^{53} - 4 q^{57} + 4 q^{59} + 8 q^{63} + 28 q^{69} - 8 q^{77} + 154 q^{81} - 20 q^{89} + 48 q^{91} - 56 q^{93} - 44 q^{97} - 8 q^{99}+O(q^{100}) $ 82 * q - 8 * q^7 - 82 * q^9 + 4 * q^13 + 4 * q^15 - 4 * q^17 - 58 * q^25 - 16 * q^29 - 12 * q^37 - 32 * q^43 + 8 * q^47 + 98 * q^49 + 6 * q^53 - 4 * q^57 + 4 * q^59 + 8 * q^63 + 28 * q^69 - 8 * q^77 + 154 * q^81 - 20 * q^89 + 48 * q^91 - 56 * q^93 - 44 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4028\mathbb{Z}\right)^\times$.

$n$	$2015$	$2281$	$2757$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	3.26219i		−	1.88343i	−0.336417	−	0.941713i	$-0.609215\pi$
$3$		−	3.26219i		−	1.88343i	0.336417	−	0.941713i	$-0.390785\pi$
$4$		0			0
$5$			2.63587i			1.17880i	0.807843	+	0.589398i	$0.200635\pi$
$5$			2.63587i			1.17880i	−0.807843	+	0.589398i	$0.799365\pi$
$6$		0			0
$7$	−3.36239			−1.27086			−0.635432	−	0.772157i	$-0.719178\pi$
$7$	−3.36239			−1.27086			−0.635432	+	0.772157i	$0.719178\pi$
$8$		0			0
$9$	−7.64188			−2.54729
$10$		0			0
$11$	−3.00331			−0.905532			−0.452766	−	0.891629i	$-0.649563\pi$
$11$	−3.00331			−0.905532			−0.452766	+	0.891629i	$0.649563\pi$
$12$		0			0
$13$	−6.81042			−1.88887			−0.944436	−	0.328697i	$-0.893391\pi$
$13$	−6.81042			−1.88887			−0.944436	+	0.328697i	$0.893391\pi$
$14$		0			0
$15$	8.59870			2.22017
$16$		0			0
$17$	−1.96976			−0.477738			−0.238869	−	0.971052i	$-0.576777\pi$
$17$	−1.96976			−0.477738			−0.238869	+	0.971052i	$0.576777\pi$
$18$		0			0
$19$			1.00000i			0.229416i
$19$			1.00000i			0.229416i
$20$		0			0
$21$			10.9687i			2.39358i
$22$		0			0
$23$		−	9.19408i		−	1.91710i	−0.284926	−	0.958550i	$-0.591969\pi$
$23$		−	9.19408i		−	1.91710i	0.284926	−	0.958550i	$-0.408031\pi$
$24$		0			0
$25$	−1.94780			−0.389560
$26$		0			0
$27$			15.1427i			2.91422i
$28$		0			0
$29$	0.0365021			0.00677828			0.00338914	−	0.999994i	$-0.498921\pi$
$29$	0.0365021			0.00677828			0.00338914	+	0.999994i	$0.498921\pi$
$30$		0			0
$31$			5.90082i			1.05982i	0.848054	+	0.529909i	$0.177774\pi$
$31$			5.90082i			1.05982i	−0.848054	+	0.529909i	$0.822226\pi$
$32$		0			0
$33$			9.79737i			1.70550i
$34$		0			0
$35$		−	8.86281i		−	1.49809i
$36$		0			0
$37$	5.76204			0.947273			0.473637	−	0.880720i	$-0.342941\pi$
$37$	5.76204			0.947273			0.473637	+	0.880720i	$0.342941\pi$
$38$		0			0
$39$			22.2169i			3.55755i
$40$		0			0
$41$		−	4.04919i		−	0.632377i	−0.948696	−	0.316188i	$-0.897597\pi$
$41$		−	4.04919i		−	0.632377i	0.948696	−	0.316188i	$-0.102403\pi$
$42$		0			0
$43$	8.69437			1.32588			0.662939	−	0.748673i	$-0.269309\pi$
$43$	8.69437			1.32588			0.662939	+	0.748673i	$0.269309\pi$
$44$		0			0
$45$		−	20.1430i		−	3.00274i
$46$		0			0
$47$	−0.478803			−0.0698406			−0.0349203	−	0.999390i	$-0.511118\pi$
$47$	−0.478803			−0.0698406			−0.0349203	+	0.999390i	$0.511118\pi$
$48$		0			0
$49$	4.30566			0.615094
$50$		0			0
$51$			6.42575i			0.899784i
$52$		0			0
$53$	−3.57810	+	6.34012i	−0.491490	+	0.870883i
$53$	−3.57810	+	6.34012i	−0.491490	+	0.870883i
$54$		0			0
$55$		−	7.91633i		−	1.06744i
$56$		0			0
$57$	3.26219			0.432088
$58$		0			0
$59$	6.03483			0.785668			0.392834	−	0.919609i	$-0.371495\pi$
$59$	6.03483			0.785668			0.392834	+	0.919609i	$0.371495\pi$
$60$		0			0
$61$		−	7.29057i		−	0.933462i	−0.884399	−	0.466731i	$-0.845432\pi$
$61$		−	7.29057i		−	0.933462i	0.884399	−	0.466731i	$-0.154568\pi$
$62$		0			0
$63$	25.6950			3.23726
$64$		0			0
$65$		−	17.9514i		−	2.22659i
$66$		0			0
$67$		−	6.45017i		−	0.788014i	−0.919107	−	0.394007i	$-0.871089\pi$
$67$		−	6.45017i		−	0.788014i	0.919107	−	0.394007i	$-0.128911\pi$
$68$		0			0
$69$	−29.9928			−3.61071
$70$		0			0
$71$		−	10.2064i		−	1.21128i	−0.795739	−	0.605640i	$-0.792917\pi$
$71$		−	10.2064i		−	1.21128i	0.795739	−	0.605640i	$-0.207083\pi$
$72$		0			0
$73$		−	5.84713i		−	0.684355i	−0.939635	−	0.342178i	$-0.888836\pi$
$73$		−	5.84713i		−	0.684355i	0.939635	−	0.342178i	$-0.111164\pi$
$74$		0			0
$75$			6.35409i			0.733707i
$76$		0			0
$77$	10.0983			1.15081
$78$		0			0
$79$			4.07524i			0.458500i	0.973368	+	0.229250i	$0.0736273\pi$
$79$			4.07524i			0.458500i	−0.973368	+	0.229250i	$0.926373\pi$
$80$		0			0
$81$	26.4727			2.94141
$82$		0			0
$83$			10.0109i			1.09884i	0.835545	+	0.549422i	$0.185152\pi$
$83$			10.0109i			1.09884i	−0.835545	+	0.549422i	$0.814848\pi$
$84$		0			0
$85$		−	5.19204i		−	0.563156i
$86$		0			0
$87$		−	0.119077i		−	0.0127664i
$88$		0			0
$89$	−16.5171			−1.75081			−0.875404	−	0.483393i	$-0.839404\pi$
$89$	−16.5171			−1.75081			−0.875404	+	0.483393i	$0.839404\pi$
$90$		0			0
$91$	22.8993			2.40050
$92$		0			0
$93$	19.2496			1.99609
$94$		0			0
$95$	−2.63587			−0.270434
$96$		0			0
$97$	−3.32473			−0.337575			−0.168788	−	0.985652i	$-0.553985\pi$
$97$	−3.32473			−0.337575			−0.168788	+	0.985652i	$0.553985\pi$
$98$		0			0
$99$	22.9510			2.30666
$100$		0			0
$101$			7.47754i			0.744043i	0.928224	+	0.372022i	$0.121335\pi$
$101$			7.47754i			0.744043i	−0.928224	+	0.372022i	$0.878665\pi$
$102$		0			0
$103$			16.2116i			1.59738i	0.601744	+	0.798689i	$0.294473\pi$
$103$			16.2116i			1.59738i	−0.601744	+	0.798689i	$0.705527\pi$
$104$		0			0
$105$	−28.9122			−2.82154
$106$		0			0
$107$	6.68786			0.646540			0.323270	−	0.946307i	$-0.395218\pi$
$107$	6.68786			0.646540			0.323270	+	0.946307i	$0.395218\pi$
$108$		0			0
$109$		−	6.79736i		−	0.651069i	−0.945530	−	0.325534i	$-0.894456\pi$
$109$		−	6.79736i		−	0.651069i	0.945530	−	0.325534i	$-0.105544\pi$
$110$		0			0
$111$		−	18.7969i		−	1.78412i
$112$		0			0
$113$	14.8582			1.39775			0.698873	−	0.715246i	$-0.253685\pi$
$113$	14.8582			1.39775			0.698873	+	0.715246i	$0.253685\pi$
$114$		0			0
$115$	24.2344			2.25987
$116$		0			0
$117$	52.0444			4.81151
$118$		0			0
$119$	6.62311			0.607140
$120$		0			0
$121$	−1.98012			−0.180011
$122$		0			0
$123$	−13.2092			−1.19104
$124$		0			0
$125$			8.04520i			0.719585i
$126$		0			0
$127$			9.89140i			0.877720i	0.898555	+	0.438860i	$0.144618\pi$
$127$			9.89140i			0.877720i	−0.898555	+	0.438860i	$0.855382\pi$
$128$		0			0
$129$		−	28.3627i		−	2.49719i
$130$		0			0
$131$	16.1220			1.40859			0.704293	−	0.709909i	$-0.251264\pi$
$131$	16.1220			1.40859			0.704293	+	0.709909i	$0.251264\pi$
$132$		0			0
$133$		−	3.36239i		−	0.291556i
$134$		0			0
$135$	−39.9142			−3.43526
$136$		0			0
$137$		−	1.07198i		−	0.0915857i	−0.998951	−	0.0457929i	$-0.985419\pi$
$137$		−	1.07198i		−	0.0915857i	0.998951	−	0.0457929i	$-0.0145814\pi$
$138$		0			0
$139$			1.40183i			0.118901i	0.998231	+	0.0594506i	$0.0189349\pi$
$139$			1.40183i			0.118901i	−0.998231	+	0.0594506i	$0.981065\pi$
$140$		0			0
$141$			1.56195i			0.131540i
$142$		0			0
$143$	20.4538			1.71043
$144$		0			0
$145$			0.0962148i			0.00799021i
$146$		0			0
$147$		−	14.0459i		−	1.15848i
$148$		0			0
$149$	−7.64644			−0.626421			−0.313211	−	0.949684i	$-0.601405\pi$
$149$	−7.64644			−0.626421			−0.313211	+	0.949684i	$0.601405\pi$
$150$		0			0
$151$		−	6.83555i		−	0.556269i	−0.960542	−	0.278135i	$-0.910284\pi$
$151$		−	6.83555i		−	0.556269i	0.960542	−	0.278135i	$-0.0897162\pi$
$152$		0			0
$153$	15.0527			1.21694
$154$		0			0
$155$	−15.5538			−1.24931
$156$		0			0
$157$			18.8646i			1.50556i	0.658273	+	0.752779i	$0.271287\pi$
$157$			18.8646i			1.50556i	−0.658273	+	0.752779i	$0.728713\pi$
$158$		0			0
$159$	20.6827	+	11.6725i	1.64024	+	0.925686i
$160$		0			0
$161$			30.9141i			2.43637i
$162$		0			0
$163$	6.33273			0.496018			0.248009	−	0.968758i	$-0.420224\pi$
$163$	6.33273			0.496018			0.248009	+	0.968758i	$0.420224\pi$
$164$		0			0
$165$	−25.8246			−2.01044
$166$		0			0
$167$			21.9292i			1.69694i	0.529247	+	0.848468i	$0.322475\pi$
$167$			21.9292i			1.69694i	−0.529247	+	0.848468i	$0.677525\pi$
$168$		0			0
$169$	33.3818			2.56783
$170$		0			0
$171$		−	7.64188i		−	0.584389i
$172$		0			0
$173$		−	8.25844i		−	0.627878i	−0.949443	−	0.313939i	$-0.898351\pi$
$173$		−	8.25844i		−	0.627878i	0.949443	−	0.313939i	$-0.101649\pi$
$174$		0			0
$175$	6.54925			0.495077
$176$		0			0
$177$		−	19.6868i		−	1.47975i
$178$		0			0
$179$		−	8.10543i		−	0.605828i	−0.953018	−	0.302914i	$-0.902040\pi$
$179$		−	8.10543i		−	0.605828i	0.953018	−	0.302914i	$-0.0979595\pi$
$180$		0			0
$181$		−	0.237186i		−	0.0176299i	−0.999961	−	0.00881494i	$-0.997194\pi$
$181$		−	0.237186i		−	0.0176299i	0.999961	−	0.00881494i	$-0.00280592\pi$
$182$		0			0
$183$	−23.7832			−1.75811
$184$		0			0
$185$			15.1880i			1.11664i
$186$		0			0
$187$	5.91582			0.432607
$188$		0			0
$189$		−	50.9157i		−	3.70357i
$190$		0			0
$191$			4.47755i			0.323984i	0.986792	+	0.161992i	$0.0517919\pi$
$191$			4.47755i			0.323984i	−0.986792	+	0.161992i	$0.948208\pi$
$192$		0			0
$193$		−	19.4545i		−	1.40037i	−0.713963	−	0.700183i	$-0.753102\pi$
$193$		−	19.4545i		−	1.40037i	0.713963	−	0.700183i	$-0.246898\pi$
$194$		0			0
$195$	−58.5608			−4.19362
$196$		0			0
$197$	24.1306			1.71924			0.859618	−	0.510937i	$-0.170702\pi$
$197$	24.1306			1.71924			0.859618	+	0.510937i	$0.170702\pi$
$198$		0			0
$199$	−19.2148			−1.36210			−0.681051	−	0.732236i	$-0.738477\pi$
$199$	−19.2148			−1.36210			−0.681051	+	0.732236i	$0.738477\pi$
$200$		0			0
$201$	−21.0417			−1.48417
$202$		0			0
$203$	−0.122734			−0.00861427
$204$		0			0
$205$	10.6731			0.745443
$206$		0			0
$207$			70.2601i			4.88342i
$208$		0			0
$209$		−	3.00331i		−	0.207743i
$210$		0			0
$211$	16.1244			1.11005			0.555025	−	0.831833i	$-0.312709\pi$
$211$	16.1244			1.11005			0.555025	+	0.831833i	$0.312709\pi$
$212$		0			0
$213$	−33.2953			−2.28136
$214$		0			0
$215$			22.9172i			1.56294i
$216$		0			0
$217$		−	19.8408i		−	1.34688i
$218$		0			0
$219$	−19.0745			−1.28893
$220$		0			0
$221$	13.4149			0.902386
$222$		0			0
$223$	−7.42165			−0.496991			−0.248495	−	0.968633i	$-0.579936\pi$
$223$	−7.42165			−0.496991			−0.248495	+	0.968633i	$0.579936\pi$
$224$		0			0
$225$	14.8848			0.992323
$226$		0			0
$227$	−28.4393			−1.88759			−0.943793	−	0.330538i	$-0.892770\pi$
$227$	−28.4393			−1.88759			−0.943793	+	0.330538i	$0.892770\pi$
$228$		0			0
$229$	−19.1138			−1.26308			−0.631539	−	0.775344i	$-0.717576\pi$
$229$	−19.1138			−1.26308			−0.631539	+	0.775344i	$0.717576\pi$
$230$		0			0
$231$		−	32.9426i		−	2.16746i
$232$		0			0
$233$			3.05315i			0.200018i	0.994987	+	0.100009i	$0.0318872\pi$
$233$			3.05315i			0.200018i	−0.994987	+	0.100009i	$0.968113\pi$
$234$		0			0
$235$		−	1.26206i		−	0.0823278i
$236$		0			0
$237$	13.2942			0.863551
$238$		0			0
$239$		−	10.2922i		−	0.665747i	−0.942972	−	0.332873i	$-0.891982\pi$
$239$		−	10.2922i		−	0.665747i	0.942972	−	0.332873i	$-0.108018\pi$
$240$		0			0
$241$	7.25837			0.467553			0.233776	−	0.972290i	$-0.424892\pi$
$241$	7.25837			0.467553			0.233776	+	0.972290i	$0.424892\pi$
$242$		0			0
$243$		−	40.9310i		−	2.62572i
$244$		0			0
$245$			11.3491i			0.725070i
$246$		0			0
$247$		−	6.81042i		−	0.433337i
$248$		0			0
$249$	32.6576			2.06959
$250$		0			0
$251$		−	15.4388i		−	0.974485i	−0.873267	−	0.487243i	$-0.838003\pi$
$251$		−	15.4388i		−	0.974485i	0.873267	−	0.487243i	$-0.161997\pi$
$252$		0			0
$253$			27.6127i			1.73600i
$254$		0			0
$255$	−16.9374			−1.06066
$256$		0			0
$257$			30.9781i			1.93236i	0.257863	+	0.966182i	$0.416982\pi$
$257$			30.9781i			1.93236i	−0.257863	+	0.966182i	$0.583018\pi$
$258$		0			0
$259$	−19.3742			−1.20385
$260$		0			0
$261$	−0.278945			−0.0172663
$262$		0			0
$263$		−	2.39203i		−	0.147499i	−0.997277	−	0.0737495i	$-0.976503\pi$
$263$		−	2.39203i		−	0.147499i	0.997277	−	0.0737495i	$-0.0234965\pi$
$264$		0			0
$265$	−16.7117	−	9.43141i	−1.02659	−	0.579367i
$266$		0			0
$267$			53.8819i			3.29752i
$268$		0			0
$269$	8.67874			0.529152			0.264576	−	0.964365i	$-0.414768\pi$
$269$	8.67874			0.529152			0.264576	+	0.964365i	$0.414768\pi$
$270$		0			0
$271$	−30.5911			−1.85828			−0.929140	−	0.369728i	$-0.879451\pi$
$271$	−30.5911			−1.85828			−0.929140	+	0.369728i	$0.879451\pi$
$272$		0			0
$273$		−	74.7018i		−	4.52116i
$274$		0			0
$275$	5.84984			0.352759
$276$		0			0
$277$			10.7612i			0.646577i	0.946300	+	0.323289i	$0.104788\pi$
$277$			10.7612i			0.646577i	−0.946300	+	0.323289i	$0.895212\pi$
$278$		0			0
$279$		−	45.0934i		−	2.69967i
$280$		0			0
$281$	−7.14502			−0.426236			−0.213118	−	0.977026i	$-0.568362\pi$
$281$	−7.14502			−0.426236			−0.213118	+	0.977026i	$0.568362\pi$
$282$		0			0
$283$			2.13870i			0.127132i	0.997978	+	0.0635661i	$0.0202474\pi$
$283$			2.13870i			0.127132i	−0.997978	+	0.0635661i	$0.979753\pi$
$284$		0			0
$285$			8.59870i			0.509343i
$286$		0			0
$287$			13.6149i			0.803665i
$288$		0			0
$289$	−13.1200			−0.771766
$290$		0			0
$291$			10.8459i			0.635798i
$292$		0			0
$293$	−20.6259			−1.20498			−0.602489	−	0.798127i	$-0.705824\pi$
$293$	−20.6259			−1.20498			−0.602489	+	0.798127i	$0.705824\pi$
$294$		0			0
$295$			15.9070i			0.926142i
$296$		0			0
$297$		−	45.4783i		−	2.63892i
$298$		0			0
$299$			62.6156i			3.62115i
$300$		0			0
$301$	−29.2338			−1.68501
$302$		0			0
$303$	24.3932			1.40135
$304$		0			0
$305$	19.2170			1.10036
$306$		0			0
$307$	17.8121			1.01659			0.508296	−	0.861182i	$-0.330275\pi$
$307$	17.8121			1.01659			0.508296	+	0.861182i	$0.330275\pi$
$308$		0			0
$309$	52.8854			3.00854
$310$		0			0
$311$	7.11341			0.403365			0.201682	−	0.979451i	$-0.435359\pi$
$311$	7.11341			0.403365			0.201682	+	0.979451i	$0.435359\pi$
$312$		0			0
$313$		−	32.8112i		−	1.85460i	−0.374320	−	0.927300i	$-0.622124\pi$
$313$		−	32.8112i		−	1.85460i	0.374320	−	0.927300i	$-0.377876\pi$
$314$		0			0
$315$			67.7286i			3.81607i
$316$		0			0
$317$	0.873145			0.0490407			0.0245204	−	0.999699i	$-0.492194\pi$
$317$	0.873145			0.0490407			0.0245204	+	0.999699i	$0.492194\pi$
$318$		0			0
$319$	−0.109627			−0.00613795
$320$		0			0
$321$		−	21.8171i		−	1.21771i
$322$		0			0
$323$		−	1.96976i		−	0.109601i
$324$		0			0
$325$	13.2653			0.735828
$326$		0			0
$327$	−22.1743			−1.22624
$328$		0			0
$329$	1.60992			0.0887578
$330$		0			0
$331$	−22.9844			−1.26334			−0.631668	−	0.775239i	$-0.717629\pi$
$331$	−22.9844			−1.26334			−0.631668	+	0.775239i	$0.717629\pi$
$332$		0			0
$333$	−44.0328			−2.41298
$334$		0			0
$335$	17.0018			0.928908
$336$		0			0
$337$		−	32.8945i		−	1.79188i	−0.444175	−	0.895940i	$-0.646503\pi$
$337$		−	32.8945i		−	1.79188i	0.444175	−	0.895940i	$-0.353497\pi$
$338$		0			0
$339$		−	48.4704i		−	2.63255i
$340$		0			0
$341$		−	17.7220i		−	0.959700i
$342$		0			0
$343$	9.05943			0.489163
$344$		0			0
$345$		−	79.0572i		−	4.25630i
$346$		0			0
$347$	23.8426			1.27994			0.639968	−	0.768402i	$-0.278948\pi$
$347$	23.8426			1.27994			0.639968	+	0.768402i	$0.278948\pi$
$348$		0			0
$349$		−	19.8638i		−	1.06328i	−0.846970	−	0.531641i	$-0.821576\pi$
$349$		−	19.8638i		−	1.06328i	0.846970	−	0.531641i	$-0.178424\pi$
$350$		0			0
$351$		−	103.128i		−	5.50458i
$352$		0			0
$353$			5.66307i			0.301415i	0.988578	+	0.150707i	$0.0481551\pi$
$353$			5.66307i			0.301415i	−0.988578	+	0.150707i	$0.951845\pi$
$354$		0			0
$355$	26.9028			1.42785
$356$		0			0
$357$		−	21.6059i		−	1.14350i
$358$		0			0
$359$			8.92323i			0.470950i	0.971880	+	0.235475i	$0.0756645\pi$
$359$			8.92323i			0.470950i	−0.971880	+	0.235475i	$0.924335\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$			6.45953i			0.339037i
$364$		0			0
$365$	15.4123			0.806715
$366$		0			0
$367$	12.3296			0.643602			0.321801	−	0.946807i	$-0.395712\pi$
$367$	12.3296			0.643602			0.321801	+	0.946807i	$0.395712\pi$
$368$		0			0
$369$			30.9434i			1.61085i
$370$		0			0
$371$	12.0310	−	21.3180i	0.624617	−	1.10677i
$372$		0			0
$373$			4.51200i			0.233623i	0.993154	+	0.116811i	$0.0372673\pi$
$373$			4.51200i			0.233623i	−0.993154	+	0.116811i	$0.962733\pi$
$374$		0			0
$375$	26.2450			1.35528
$376$		0			0
$377$	−0.248595			−0.0128033
$378$		0			0
$379$			15.3097i			0.786405i	0.919452	+	0.393202i	$0.128633\pi$
$379$			15.3097i			0.786405i	−0.919452	+	0.393202i	$0.871367\pi$
$380$		0			0
$381$	32.2676			1.65312
$382$		0			0
$383$			19.7415i			1.00875i	0.863486	+	0.504373i	$0.168276\pi$
$383$			19.7415i			1.00875i	−0.863486	+	0.504373i	$0.831724\pi$
$384$		0			0
$385$			26.6178i			1.35657i
$386$		0			0
$387$	−66.4413			−3.37740
$388$		0			0
$389$			5.43932i			0.275785i	0.990447	+	0.137892i	$0.0440328\pi$
$389$			5.43932i			0.275785i	−0.990447	+	0.137892i	$0.955967\pi$
$390$		0			0
$391$			18.1102i			0.915871i
$392$		0			0
$393$		−	52.5931i		−	2.65297i
$394$		0			0
$395$	−10.7418			−0.540478
$396$		0			0
$397$			19.1476i			0.960989i	0.876998	+	0.480495i	$0.159543\pi$
$397$			19.1476i			0.960989i	−0.876998	+	0.480495i	$0.840457\pi$
$398$		0			0
$399$	−10.9687			−0.549124
$400$		0			0
$401$			11.0769i			0.553154i	0.960992	+	0.276577i	$0.0892000\pi$
$401$			11.0769i			0.553154i	−0.960992	+	0.276577i	$0.910800\pi$
$402$		0			0
$403$		−	40.1871i		−	2.00186i
$404$		0			0
$405$			69.7786i			3.46733i
$406$		0			0
$407$	−17.3052			−0.857787
$408$		0			0
$409$	−17.4367			−0.862187			−0.431094	−	0.902307i	$-0.641872\pi$
$409$	−17.4367			−0.862187			−0.431094	+	0.902307i	$0.641872\pi$
$410$		0			0
$411$	−3.49701			−0.172495
$412$		0			0
$413$	−20.2914			−0.998477
$414$		0			0
$415$	−26.3875			−1.29531
$416$		0			0
$417$	4.57302			0.223942
$418$		0			0
$419$		−	35.7483i		−	1.74642i	−0.487343	−	0.873210i	$-0.662034\pi$
$419$		−	35.7483i		−	1.74642i	0.487343	−	0.873210i	$-0.337966\pi$
$420$		0			0
$421$		−	20.1579i		−	0.982434i	−0.871037	−	0.491217i	$-0.836552\pi$
$421$		−	20.1579i		−	0.982434i	0.871037	−	0.491217i	$-0.163448\pi$
$422$		0			0
$423$	3.65896			0.177905
$424$		0			0
$425$	3.83670			0.186107
$426$		0			0
$427$			24.5137i			1.18630i
$428$		0			0
$429$		−	66.7242i		−	3.22148i
$430$		0			0
$431$	20.8441			1.00402			0.502012	−	0.864861i	$-0.332593\pi$
$431$	20.8441			1.00402			0.502012	+	0.864861i	$0.332593\pi$
$432$		0			0
$433$	18.3047			0.879667			0.439833	−	0.898079i	$-0.355038\pi$
$433$	18.3047			0.879667			0.439833	+	0.898079i	$0.355038\pi$
$434$		0			0
$435$	0.313871			0.0150490
$436$		0			0
$437$	9.19408			0.439813
$438$		0			0
$439$	32.0780			1.53100			0.765499	−	0.643437i	$-0.222492\pi$
$439$	32.0780			1.53100			0.765499	+	0.643437i	$0.222492\pi$
$440$		0			0
$441$	−32.9033			−1.56682
$442$		0			0
$443$		−	14.3707i		−	0.682773i	−0.939923	−	0.341387i	$-0.889103\pi$
$443$		−	14.3707i		−	0.682773i	0.939923	−	0.341387i	$-0.110897\pi$
$444$		0			0
$445$		−	43.5368i		−	2.06384i
$446$		0			0
$447$			24.9442i			1.17982i
$448$		0			0
$449$	32.5566			1.53644			0.768220	−	0.640185i	$-0.221143\pi$
$449$	32.5566			1.53644			0.768220	+	0.640185i	$0.221143\pi$
$450$		0			0
$451$			12.1610i			0.572638i
$452$		0			0
$453$	−22.2989			−1.04769
$454$		0			0
$455$			60.3595i			2.82970i
$456$		0			0
$457$			34.3236i			1.60559i	0.596255	+	0.802795i	$0.296655\pi$
$457$			34.3236i			1.60559i	−0.596255	+	0.802795i	$0.703345\pi$
$458$		0			0
$459$		−	29.8276i		−	1.39223i
$460$		0			0
$461$	−24.1463			−1.12460			−0.562302	−	0.826932i	$-0.690084\pi$
$461$	−24.1463			−1.12460			−0.562302	+	0.826932i	$0.690084\pi$
$462$		0			0
$463$			28.3155i			1.31593i	0.753048	+	0.657966i	$0.228583\pi$
$463$			28.3155i			1.31593i	−0.753048	+	0.657966i	$0.771417\pi$
$464$		0			0
$465$			50.7394i			2.35298i
$466$		0			0
$467$	30.4315			1.40820			0.704100	−	0.710100i	$-0.251350\pi$
$467$	30.4315			1.40820			0.704100	+	0.710100i	$0.251350\pi$
$468$		0			0
$469$			21.6880i			1.00146i
$470$		0			0
$471$	61.5399			2.83561
$472$		0			0
$473$	−26.1119			−1.20063
$474$		0			0
$475$		−	1.94780i		−	0.0893711i
$476$		0			0
$477$	27.3435	−	48.4505i	1.25197	−	2.21840i
$478$		0			0
$479$		−	6.81523i		−	0.311396i	−0.987805	−	0.155698i	$-0.950237\pi$
$479$		−	6.81523i		−	0.311396i	0.987805	−	0.155698i	$-0.0497627\pi$
$480$		0			0
$481$	−39.2419			−1.78928
$482$		0			0
$483$	100.848			4.58873
$484$		0			0
$485$		−	8.76355i		−	0.397932i
$486$		0			0
$487$	9.67812			0.438558			0.219279	−	0.975662i	$-0.429630\pi$
$487$	9.67812			0.438558			0.219279	+	0.975662i	$0.429630\pi$
$488$		0			0
$489$		−	20.6586i		−	0.934212i
$490$		0			0
$491$		−	4.69657i		−	0.211953i	−0.994369	−	0.105977i	$-0.966203\pi$
$491$		−	4.69657i		−	0.211953i	0.994369	−	0.105977i	$-0.0337969\pi$
$492$		0			0
$493$	−0.0719006			−0.00323824
$494$		0			0
$495$			60.4957i			2.71908i
$496$		0			0
$497$			34.3180i			1.53937i
$498$		0			0
$499$			31.7460i			1.42114i	0.703624	+	0.710572i	$0.251564\pi$
$499$			31.7460i			1.42114i	−0.703624	+	0.710572i	$0.748436\pi$
$500$		0			0
$501$	71.5374			3.19605
$502$		0			0
$503$		−	22.8440i		−	1.01856i	−0.860600	−	0.509281i	$-0.829911\pi$
$503$		−	22.8440i		−	1.01856i	0.860600	−	0.509281i	$-0.170089\pi$
$504$		0			0
$505$	−19.7098			−0.877075
$506$		0			0
$507$		−	108.898i		−	4.83633i
$508$		0			0
$509$		−	12.4104i		−	0.550081i	−0.961433	−	0.275040i	$-0.911309\pi$
$509$		−	12.4104i		−	0.550081i	0.961433	−	0.275040i	$-0.0886913\pi$
$510$		0			0
$511$			19.6603i			0.869722i
$512$		0			0
$513$	−15.1427			−0.668567
$514$		0			0
$515$	−42.7317			−1.88298
$516$		0			0
$517$	1.43799			0.0632429
$518$		0			0
$519$	−26.9406			−1.18256
$520$		0			0
$521$	33.2019			1.45460			0.727301	−	0.686319i	$-0.240774\pi$
$521$	33.2019			1.45460			0.727301	+	0.686319i	$0.240774\pi$
$522$		0			0
$523$	6.48458			0.283551			0.141775	−	0.989899i	$-0.454719\pi$
$523$	6.48458			0.283551			0.141775	+	0.989899i	$0.454719\pi$
$524$		0			0
$525$		−	21.3649i		−	0.932441i
$526$		0			0
$527$		−	11.6232i		−	0.506315i
$528$		0			0
$529$	−61.5312			−2.67527
$530$		0			0
$531$	−46.1175			−2.00133
$532$		0			0
$533$			27.5767i			1.19448i
$534$		0			0
$535$			17.6283i			0.762139i
$536$		0			0
$537$	−26.4415			−1.14103
$538$		0			0
$539$	−12.9312			−0.556987
$540$		0			0
$541$	−10.8539			−0.466645			−0.233322	−	0.972399i	$-0.574960\pi$
$541$	−10.8539			−0.466645			−0.233322	+	0.972399i	$0.574960\pi$
$542$		0			0
$543$	−0.773745			−0.0332046
$544$		0			0
$545$	17.9169			0.767477
$546$		0			0
$547$	−25.6085			−1.09494			−0.547471	−	0.836825i	$-0.684409\pi$
$547$	−25.6085			−1.09494			−0.547471	+	0.836825i	$0.684409\pi$
$548$		0			0
$549$			55.7137i			2.37780i
$550$		0			0
$551$			0.0365021i			0.00155504i
$552$		0			0
$553$		−	13.7025i		−	0.582691i
$554$		0			0
$555$	49.5460			2.10311
$556$		0			0
$557$		−	13.5656i		−	0.574795i	−0.957811	−	0.287397i	$-0.907210\pi$
$557$		−	13.5656i		−	0.574795i	0.957811	−	0.287397i	$-0.0927900\pi$
$558$		0			0
$559$	−59.2123			−2.50441
$560$		0			0
$561$		−	19.2985i		−	0.814784i
$562$		0			0
$563$		−	7.84717i		−	0.330719i	−0.986233	−	0.165359i	$-0.947122\pi$
$563$		−	7.84717i		−	0.330719i	0.986233	−	0.165359i	$-0.0528784\pi$
$564$		0			0
$565$			39.1644i			1.64766i
$566$		0			0
$567$	−89.0116			−3.73814
$568$		0			0
$569$			22.3303i			0.936137i	0.883692	+	0.468068i	$0.155050\pi$
$569$			22.3303i			0.936137i	−0.883692	+	0.468068i	$0.844950\pi$
$570$		0			0
$571$		−	15.4020i		−	0.644554i	−0.946645	−	0.322277i	$-0.895552\pi$
$571$		−	15.4020i		−	0.644554i	0.946645	−	0.322277i	$-0.104448\pi$
$572$		0			0
$573$	14.6066			0.610200
$574$		0			0
$575$			17.9082i			0.746824i
$576$		0			0
$577$	41.7571			1.73837			0.869186	−	0.494485i	$-0.164643\pi$
$577$	41.7571			1.73837			0.869186	+	0.494485i	$0.164643\pi$
$578$		0			0
$579$	−63.4643			−2.63749
$580$		0			0
$581$		−	33.6607i		−	1.39648i
$582$		0			0
$583$	10.7462	−	19.0414i	0.445060	−	0.788613i
$584$		0			0
$585$			137.182i			5.67179i
$586$		0			0
$587$	8.57233			0.353818			0.176909	−	0.984227i	$-0.443390\pi$
$587$	8.57233			0.353818			0.176909	+	0.984227i	$0.443390\pi$
$588$		0			0
$589$	−5.90082			−0.243139
$590$		0			0
$591$		−	78.7187i		−	3.23805i
$592$		0			0
$593$	−27.7214			−1.13838			−0.569190	−	0.822206i	$-0.692743\pi$
$593$	−27.7214			−1.13838			−0.569190	+	0.822206i	$0.692743\pi$
$594$		0			0
$595$			17.4576i			0.715694i
$596$		0			0
$597$			62.6824i			2.56542i
$598$		0			0
$599$	2.77624			0.113434			0.0567170	−	0.998390i	$-0.481937\pi$
$599$	2.77624			0.113434			0.0567170	+	0.998390i	$0.481937\pi$
$600$		0			0
$601$			13.7025i			0.558935i	0.960155	+	0.279468i	$0.0901580\pi$
$601$			13.7025i			0.558935i	−0.960155	+	0.279468i	$0.909842\pi$
$602$		0			0
$603$			49.2915i			2.00730i
$604$		0			0
$605$		−	5.21934i		−	0.212196i
$606$		0			0
$607$	−9.31736			−0.378180			−0.189090	−	0.981960i	$-0.560554\pi$
$607$	−9.31736			−0.378180			−0.189090	+	0.981960i	$0.560554\pi$
$608$		0			0
$609$			0.400383i			0.0162243i
$610$		0			0
$611$	3.26085			0.131920
$612$		0			0
$613$			25.8232i			1.04299i	0.853255	+	0.521494i	$0.174625\pi$
$613$			25.8232i			1.04299i	−0.853255	+	0.521494i	$0.825375\pi$
$614$		0			0
$615$		−	34.8178i		−	1.40399i
$616$		0			0
$617$			27.8017i			1.11925i	0.828744	+	0.559627i	$0.189056\pi$
$617$			27.8017i			1.11925i	−0.828744	+	0.559627i	$0.810944\pi$
$618$		0			0
$619$	−11.7380			−0.471789			−0.235894	−	0.971779i	$-0.575802\pi$
$619$	−11.7380			−0.471789			−0.235894	+	0.971779i	$0.575802\pi$
$620$		0			0
$621$	139.223			5.58684
$622$		0			0
$623$	55.5369			2.22504
$624$		0			0
$625$	−30.9451			−1.23780
$626$		0			0
$627$	−9.79737			−0.391269
$628$		0			0
$629$	−11.3499			−0.452548
$630$		0			0
$631$			39.7778i			1.58353i	0.610827	+	0.791764i	$0.290837\pi$
$631$			39.7778i			1.58353i	−0.610827	+	0.791764i	$0.709163\pi$
$632$		0			0
$633$		−	52.6009i		−	2.09070i
$634$		0			0
$635$	−26.0724			−1.03465
$636$		0			0
$637$	−29.3233			−1.16183
$638$		0			0
$639$			77.9963i			3.08549i
$640$		0			0
$641$			12.2389i			0.483409i	0.970350	+	0.241704i	$0.0777064\pi$
$641$			12.2389i			0.483409i	−0.970350	+	0.241704i	$0.922294\pi$
$642$		0			0
$643$	−7.62705			−0.300781			−0.150391	−	0.988627i	$-0.548053\pi$
$643$	−7.62705			−0.300781			−0.150391	+	0.988627i	$0.548053\pi$
$644$		0			0
$645$	74.7603			2.94368
$646$		0			0
$647$	40.0892			1.57607			0.788035	−	0.615630i	$-0.211098\pi$
$647$	40.0892			1.57607			0.788035	+	0.615630i	$0.211098\pi$
$648$		0			0
$649$	−18.1245			−0.711448
$650$		0			0
$651$	−64.7246			−2.53676
$652$		0			0
$653$	−38.1919			−1.49457			−0.747283	−	0.664506i	$-0.768642\pi$
$653$	−38.1919			−1.49457			−0.747283	+	0.664506i	$0.768642\pi$
$654$		0			0
$655$			42.4955i			1.66044i
$656$		0			0
$657$			44.6831i			1.74325i
$658$		0			0
$659$		−	21.4762i		−	0.836594i	−0.908310	−	0.418297i	$-0.862627\pi$
$659$		−	21.4762i		−	0.836594i	0.908310	−	0.418297i	$-0.137373\pi$
$660$		0			0
$661$	22.8830			0.890047			0.445024	−	0.895519i	$-0.353195\pi$
$661$	22.8830			0.890047			0.445024	+	0.895519i	$0.353195\pi$
$662$		0			0
$663$		−	43.7620i		−	1.69958i
$664$		0			0
$665$	8.86281			0.343685
$666$		0			0
$667$		−	0.335604i		−	0.0129946i
$668$		0			0
$669$			24.2108i			0.936045i
$670$		0			0
$671$			21.8959i			0.845280i
$672$		0			0
$673$	23.9516			0.923268			0.461634	−	0.887070i	$-0.347263\pi$
$673$	23.9516			0.923268			0.461634	+	0.887070i	$0.347263\pi$
$674$		0			0
$675$		−	29.4949i		−	1.13526i
$676$		0			0
$677$		−	13.1771i		−	0.506437i	−0.967409	−	0.253219i	$-0.918511\pi$
$677$		−	13.1771i		−	0.506437i	0.967409	−	0.253219i	$-0.0814892\pi$
$678$		0			0
$679$	11.1790			0.429012
$680$		0			0
$681$			92.7745i			3.55513i
$682$		0			0
$683$	−34.6166			−1.32457			−0.662283	−	0.749254i	$-0.730412\pi$
$683$	−34.6166			−1.32457			−0.662283	+	0.749254i	$0.730412\pi$
$684$		0			0
$685$	2.82561			0.107961
$686$		0			0
$687$			62.3529i			2.37891i
$688$		0			0
$689$	24.3684	−	43.1789i	0.928362	−	1.64499i
$690$		0			0
$691$		−	20.8061i		−	0.791503i	−0.918358	−	0.395752i	$-0.870484\pi$
$691$		−	20.8061i		−	0.791503i	0.918358	−	0.395752i	$-0.129516\pi$
$692$		0			0
$693$	−77.1700			−2.93145
$694$		0			0
$695$	−3.69503			−0.140160
$696$		0			0
$697$			7.97595i			0.302111i
$698$		0			0
$699$	9.95995			0.376720
$700$		0			0
$701$		−	41.6830i		−	1.57434i	−0.616733	−	0.787172i	$-0.711544\pi$
$701$		−	41.6830i		−	1.57434i	0.616733	−	0.787172i	$-0.288456\pi$
$702$		0			0
$703$			5.76204i			0.217319i
$704$		0			0
$705$	−4.11708			−0.155058
$706$		0			0
$707$		−	25.1424i		−	0.945577i
$708$		0			0
$709$			47.6858i			1.79088i	0.445185	+	0.895439i	$0.353138\pi$
$709$			47.6858i			1.79088i	−0.445185	+	0.895439i	$0.646862\pi$
$710$		0			0
$711$		−	31.1425i		−	1.16793i
$712$		0			0
$713$	54.2526			2.03178
$714$		0			0
$715$			53.9136i			2.01625i
$716$		0			0
$717$	−33.5751			−1.25388
$718$		0			0
$719$			24.0724i			0.897750i	0.893595	+	0.448875i	$0.148175\pi$
$719$			24.0724i			0.897750i	−0.893595	+	0.448875i	$0.851825\pi$
$720$		0			0
$721$		−	54.5098i		−	2.03005i
$722$		0			0
$723$		−	23.6782i		−	0.880602i
$724$		0			0
$725$	−0.0710988			−0.00264054
$726$		0			0
$727$	20.9122			0.775590			0.387795	−	0.921746i	$-0.373237\pi$
$727$	20.9122			0.775590			0.387795	+	0.921746i	$0.373237\pi$
$728$		0			0
$729$	−54.1064			−2.00394
$730$		0			0
$731$	−17.1259			−0.633423
$732$		0			0
$733$	6.42996			0.237496			0.118748	−	0.992924i	$-0.462112\pi$
$733$	6.42996			0.237496			0.118748	+	0.992924i	$0.462112\pi$
$734$		0			0
$735$	37.0230			1.36562
$736$		0			0
$737$			19.3719i			0.713572i
$738$		0			0
$739$		−	17.0182i		−	0.626025i	−0.949749	−	0.313013i	$-0.898662\pi$
$739$		−	17.0182i		−	0.626025i	0.949749	−	0.313013i	$-0.101338\pi$
$740$		0			0
$741$	−22.2169			−0.816158
$742$		0			0
$743$	23.5885			0.865380			0.432690	−	0.901543i	$-0.357565\pi$
$743$	23.5885			0.865380			0.432690	+	0.901543i	$0.357565\pi$
$744$		0			0
$745$		−	20.1550i		−	0.738423i
$746$		0			0
$747$		−	76.5025i		−	2.79908i
$748$		0			0
$749$	−22.4872			−0.821664
$750$		0			0
$751$	25.5160			0.931092			0.465546	−	0.885024i	$-0.345858\pi$
$751$	25.5160			0.931092			0.465546	+	0.885024i	$0.345858\pi$
$752$		0			0
$753$	−50.3641			−1.83537
$754$		0			0
$755$	18.0176			0.655728
$756$		0			0
$757$	40.2602			1.46328			0.731641	−	0.681690i	$-0.238755\pi$
$757$	40.2602			1.46328			0.731641	+	0.681690i	$0.238755\pi$
$758$		0			0
$759$	90.0779			3.26962
$760$		0			0
$761$			25.2159i			0.914074i	0.889448	+	0.457037i	$0.151089\pi$
$761$			25.2159i			0.914074i	−0.889448	+	0.457037i	$0.848911\pi$
$762$		0			0
$763$			22.8554i			0.827420i
$764$		0			0
$765$			39.6769i			1.43452i
$766$		0			0
$767$	−41.0997			−1.48403
$768$		0			0
$769$		−	4.97540i		−	0.179418i	−0.995968	−	0.0897088i	$-0.971406\pi$
$769$		−	4.97540i		−	0.179418i	0.995968	−	0.0897088i	$-0.0285936\pi$
$770$		0			0
$771$	101.057			3.63946
$772$		0			0
$773$		−	36.6087i		−	1.31672i	−0.752702	−	0.658362i	$-0.771250\pi$
$773$		−	36.6087i		−	1.31672i	0.752702	−	0.658362i	$-0.228750\pi$
$774$		0			0
$775$		−	11.4936i		−	0.412862i
$776$		0			0
$777$			63.2024i			2.26737i
$778$		0			0
$779$	4.04919			0.145077
$780$		0			0
$781$			30.6531i			1.09685i
$782$		0			0
$783$			0.552741i			0.0197534i
$784$		0			0
$785$	−49.7246			−1.77475
$786$		0			0
$787$		−	19.4012i		−	0.691578i	−0.938312	−	0.345789i	$-0.887611\pi$
$787$		−	19.4012i		−	0.691578i	0.938312	−	0.345789i	$-0.112389\pi$
$788$		0			0
$789$	−7.80326			−0.277804
$790$		0			0
$791$	−49.9592			−1.77634
$792$		0			0
$793$			49.6519i			1.76319i
$794$		0			0
$795$	−30.7670	+	54.5168i	−1.09119	+	1.93351i
$796$		0			0
$797$		−	7.38110i		−	0.261452i	−0.991419	−	0.130726i	$-0.958269\pi$
$797$		−	7.38110i		−	0.261452i	0.991419	−	0.130726i	$-0.0417308\pi$
$798$		0			0
$799$	0.943129			0.0333655
$800$		0			0
$801$	126.222			4.45982
$802$		0			0
$803$			17.5608i			0.619706i
$804$		0			0
$805$	−81.4854			−2.87198
$806$		0			0
$807$		−	28.3117i		−	0.996619i
$808$		0			0
$809$			4.09097i			0.143831i	0.997411	+	0.0719155i	$0.0229112\pi$
$809$			4.09097i			0.143831i	−0.997411	+	0.0719155i	$0.977089\pi$
$810$		0			0
$811$	2.25553			0.0792023			0.0396011	−	0.999216i	$-0.487391\pi$
$811$	2.25553			0.0792023			0.0396011	+	0.999216i	$0.487391\pi$
$812$		0			0
$813$			99.7941i			3.49993i
$814$		0			0
$815$			16.6922i			0.584703i
$816$		0			0
$817$			8.69437i			0.304177i
$818$		0			0
$819$	−174.994			−6.11477
$820$		0			0
$821$			26.6265i			0.929272i	0.885502	+	0.464636i	$0.153815\pi$
$821$			26.6265i			0.929272i	−0.885502	+	0.464636i	$0.846185\pi$
$822$		0			0
$823$	45.9165			1.60055			0.800274	−	0.599635i	$-0.204687\pi$
$823$	45.9165			1.60055			0.800274	+	0.599635i	$0.204687\pi$
$824$		0			0
$825$		−	19.0833i		−	0.664395i
$826$		0			0
$827$		−	11.0087i		−	0.382810i	−0.981511	−	0.191405i	$-0.938696\pi$
$827$		−	11.0087i		−	0.382810i	0.981511	−	0.191405i	$-0.0613044\pi$
$828$		0			0
$829$		−	0.174469i		−	0.00605956i	−0.999995	−	0.00302978i	$-0.999036\pi$
$829$		−	0.174469i		−	0.00605956i	0.999995	−	0.00302978i	$-0.000964411\pi$
$830$		0			0
$831$	35.1050			1.21778
$832$		0			0
$833$	−8.48113			−0.293854
$834$		0			0
$835$	−57.8026			−2.00034
$836$		0			0
$837$	−89.3543			−3.08854
$838$		0			0
$839$	2.30009			0.0794078			0.0397039	−	0.999211i	$-0.487359\pi$
$839$	2.30009			0.0794078			0.0397039	+	0.999211i	$0.487359\pi$
$840$		0			0
$841$	−28.9987			−0.999954
$842$		0			0
$843$			23.3084i			0.802784i
$844$		0			0
$845$			87.9901i			3.02695i
$846$		0			0
$847$	6.65794			0.228769
$848$		0			0
$849$	6.97683			0.239444
$850$		0			0
$851$		−	52.9767i		−	1.81602i
$852$		0			0
$853$		−	32.5869i		−	1.11576i	−0.829923	−	0.557878i	$-0.811616\pi$
$853$		−	32.5869i		−	1.11576i	0.829923	−	0.557878i	$-0.188384\pi$
$854$		0			0
$855$	20.1430			0.688876
$856$		0			0
$857$	−3.62671			−0.123886			−0.0619430	−	0.998080i	$-0.519730\pi$
$857$	−3.62671			−0.123886			−0.0619430	+	0.998080i	$0.519730\pi$
$858$		0			0
$859$	4.43073			0.151175			0.0755873	−	0.997139i	$-0.475917\pi$
$859$	4.43073			0.151175			0.0755873	+	0.997139i	$0.475917\pi$
$860$		0			0
$861$	44.4145			1.51364
$862$		0			0
$863$	42.2126			1.43693			0.718467	−	0.695561i	$-0.244844\pi$
$863$	42.2126			1.43693			0.718467	+	0.695561i	$0.244844\pi$
$864$		0			0
$865$	21.7682			0.740140
$866$		0			0
$867$			42.8000i			1.45357i
$868$		0			0
$869$		−	12.2392i		−	0.415187i
$870$		0			0
$871$			43.9284i			1.48846i
$872$		0			0
$873$	25.4072			0.859904
$874$		0			0
$875$		−	27.0511i		−	0.914494i
$876$		0			0
$877$	25.6294			0.865443			0.432721	−	0.901528i	$-0.357553\pi$
$877$	25.6294			0.865443			0.432721	+	0.901528i	$0.357553\pi$
$878$		0			0
$879$			67.2856i			2.26949i
$880$		0			0
$881$			41.4991i			1.39814i	0.715052	+	0.699071i	$0.246403\pi$
$881$			41.4991i			1.39814i	−0.715052	+	0.699071i	$0.753597\pi$
$882$		0			0
$883$			7.43053i			0.250057i	0.992153	+	0.125029i	$0.0399023\pi$
$883$			7.43053i			0.250057i	−0.992153	+	0.125029i	$0.960098\pi$
$884$		0			0
$885$	51.8917			1.74432
$886$		0			0
$887$			18.7746i			0.630389i	0.949027	+	0.315194i	$0.102070\pi$
$887$			18.7746i			0.630389i	−0.949027	+	0.315194i	$0.897930\pi$
$888$		0			0
$889$		−	33.2587i		−	1.11546i
$890$		0			0
$891$	−79.5059			−2.66355
$892$		0			0
$893$		−	0.478803i		−	0.0160225i
$894$		0			0
$895$	21.3648			0.714148
$896$		0			0
$897$	204.264			6.82017
$898$		0			0
$899$			0.215392i			0.00718374i
$900$		0			0
$901$	7.04802	−	12.4886i	0.234804	−	0.416054i
$902$		0			0
$903$			95.3663i			3.17359i
$904$		0			0
$905$	0.625190			0.0207820
$906$		0			0
$907$	−41.5635			−1.38009			−0.690047	−	0.723764i	$-0.742410\pi$
$907$	−41.5635			−1.38009			−0.690047	+	0.723764i	$0.742410\pi$
$908$		0			0
$909$		−	57.1425i		−	1.89530i
$910$		0			0
$911$	17.7741			0.588881			0.294441	−	0.955670i	$-0.404867\pi$
$911$	17.7741			0.588881			0.294441	+	0.955670i	$0.404867\pi$
$912$		0			0
$913$		−	30.0660i		−	0.995039i
$914$		0			0
$915$		−	62.6894i		−	2.07245i
$916$		0			0
$917$	−54.2085			−1.79012
$918$		0			0
$919$		−	0.540757i		−	0.0178379i	−0.999960	−	0.00891896i	$-0.997161\pi$
$919$		−	0.540757i		−	0.0178379i	0.999960	−	0.00891896i	$-0.00283903\pi$
$920$		0			0
$921$		−	58.1066i		−	1.91468i
$922$		0			0
$923$			69.5100i			2.28795i
$924$		0			0
$925$	−11.2233			−0.369019
$926$		0			0
$927$		−	123.887i		−	4.06899i
$928$		0			0
$929$	−25.5323			−0.837688			−0.418844	−	0.908058i	$-0.637565\pi$
$929$	−25.5323			−0.837688			−0.418844	+	0.908058i	$0.637565\pi$
$930$		0			0
$931$			4.30566i			0.141112i
$932$		0			0
$933$		−	23.2053i		−	0.759707i
$934$		0			0
$935$			15.5933i			0.509956i
$936$		0			0
$937$	41.8306			1.36655			0.683273	−	0.730163i	$-0.260556\pi$
$937$	41.8306			1.36655			0.683273	+	0.730163i	$0.260556\pi$
$938$		0			0
$939$	−107.036			−3.49300
$940$		0			0
$941$	−32.7771			−1.06850			−0.534251	−	0.845326i	$-0.679406\pi$
$941$	−32.7771			−1.06850			−0.534251	+	0.845326i	$0.679406\pi$
$942$		0			0
$943$	−37.2286			−1.21233
$944$		0			0
$945$	134.207			4.36575
$946$		0			0
$947$	31.0839			1.01009			0.505046	−	0.863093i	$-0.331476\pi$
$947$	31.0839			1.01009			0.505046	+	0.863093i	$0.331476\pi$
$948$		0			0
$949$			39.8214i			1.29266i
$950$		0			0
$951$		−	2.84837i		−	0.0923646i
$952$		0			0
$953$	5.23757			0.169662			0.0848308	−	0.996395i	$-0.472965\pi$
$953$	5.23757			0.169662			0.0848308	+	0.996395i	$0.472965\pi$
$954$		0			0
$955$	−11.8022			−0.381911
$956$		0			0
$957$			0.357625i			0.0115604i
$958$		0			0
$959$			3.60443i			0.116393i
$960$		0			0
$961$	−3.81965			−0.123214
$962$		0			0
$963$	−51.1079			−1.64693
$964$		0			0
$965$	51.2795			1.65075
$966$		0			0
$967$	13.1527			0.422962			0.211481	−	0.977382i	$-0.432171\pi$
$967$	13.1527			0.422962			0.211481	+	0.977382i	$0.432171\pi$
$968$		0			0
$969$	−6.42575			−0.206425
$970$		0			0
$971$	5.69636			0.182805			0.0914024	−	0.995814i	$-0.470865\pi$
$971$	5.69636			0.182805			0.0914024	+	0.995814i	$0.470865\pi$
$972$		0			0
$973$		−	4.71348i		−	0.151107i
$974$		0			0
$975$		−	43.2740i		−	1.38588i
$976$		0			0
$977$		−	4.36943i		−	0.139790i	−0.997554	−	0.0698952i	$-0.977734\pi$
$977$		−	4.36943i		−	0.139790i	0.997554	−	0.0698952i	$-0.0222665\pi$
$978$		0			0
$979$	49.6059			1.58541
$980$		0			0
$981$			51.9446i			1.65846i
$982$		0			0
$983$	47.0728			1.50139			0.750694	−	0.660650i	$-0.229719\pi$
$983$	47.0728			1.50139			0.750694	+	0.660650i	$0.229719\pi$
$984$		0			0
$985$			63.6051i			2.02663i
$986$		0			0
$987$		−	5.25187i		−	0.167169i
$988$		0			0
$989$		−	79.9368i		−	2.54184i
$990$		0			0
$991$	−27.8568			−0.884900			−0.442450	−	0.896793i	$-0.645891\pi$
$991$	−27.8568			−0.884900			−0.442450	+	0.896793i	$0.645891\pi$
$992$		0			0
$993$			74.9794i			2.37940i
$994$		0			0
$995$		−	50.6477i		−	1.60564i
$996$		0			0
$997$	−19.3597			−0.613129			−0.306565	−	0.951850i	$-0.599180\pi$
$997$	−19.3597			−0.613129			−0.306565	+	0.951850i	$0.599180\pi$
$998$		0			0
$999$			87.2529i			2.76056i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.c.a.3497.3	✓	82
53.52	even	2	inner	4028.2.c.a.3497.80	yes	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.c.a.3497.3	✓	82	1.1	even	1	trivial
4028.2.c.a.3497.80	yes	82	53.52	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4028.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.26219i q^{3} +2.63587i q^{5} -3.36239 q^{7} -7.64188 q^{9} +O(q^{10})\)	\(q-3.26219i q^{3} +2.63587i q^{5} -3.36239 q^{7} -7.64188 q^{9} -3.00331 q^{11} -6.81042 q^{13} +8.59870 q^{15} -1.96976 q^{17} +1.00000i q^{19} +10.9687i q^{21} -9.19408i q^{23} -1.94780 q^{25} +15.1427i q^{27} +0.0365021 q^{29} +5.90082i q^{31} +9.79737i q^{33} -8.86281i q^{35} +5.76204 q^{37} +22.2169i q^{39} -4.04919i q^{41} +8.69437 q^{43} -20.1430i q^{45} -0.478803 q^{47} +4.30566 q^{49} +6.42575i q^{51} +(-3.57810 + 6.34012i) q^{53} -7.91633i q^{55} +3.26219 q^{57} +6.03483 q^{59} -7.29057i q^{61} +25.6950 q^{63} -17.9514i q^{65} -6.45017i q^{67} -29.9928 q^{69} -10.2064i q^{71} -5.84713i q^{73} +6.35409i q^{75} +10.0983 q^{77} +4.07524i q^{79} +26.4727 q^{81} +10.0109i q^{83} -5.19204i q^{85} -0.119077i q^{87} -16.5171 q^{89} +22.8993 q^{91} +19.2496 q^{93} -2.63587 q^{95} -3.32473 q^{97} +22.9510 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 8 q^{7} - 82 q^{9}+O(q^{10}) \) 82 * q - 8 * q^7 - 82 * q^9	\( 82 q - 8 q^{7} - 82 q^{9} + 4 q^{13} + 4 q^{15} - 4 q^{17} - 58 q^{25} - 16 q^{29} - 12 q^{37} - 32 q^{43} + 8 q^{47} + 98 q^{49} + 6 q^{53} - 4 q^{57} + 4 q^{59} + 8 q^{63} + 28 q^{69} - 8 q^{77} + 154 q^{81} - 20 q^{89} + 48 q^{91} - 56 q^{93} - 44 q^{97} - 8 q^{99}+O(q^{100}) \) 82 * q - 8 * q^7 - 82 * q^9 + 4 * q^13 + 4 * q^15 - 4 * q^17 - 58 * q^25 - 16 * q^29 - 12 * q^37 - 32 * q^43 + 8 * q^47 + 98 * q^49 + 6 * q^53 - 4 * q^57 + 4 * q^59 + 8 * q^63 + 28 * q^69 - 8 * q^77 + 154 * q^81 - 20 * q^89 + 48 * q^91 - 56 * q^93 - 44 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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